The study finds non-uniform lattices with thin Hitchin representations in specific Lie groups.
problem Finding thin Hitchin representations in non-uniform lattices of Lie groups.
method Arithmetic methods to construct thin Hitchin representations.
result Infinitely many orbits of thin Hitchin representations in non-uniform lattices.
The study finds that certain hyperbolic manifolds contain subgroups isomorphic to surface groups.
problem The existence of thin surface subgroups in non-uniform arithmetic lattices.
method Analyzes arithmetic hyperbolic manifolds and their fundamental groups.
result Fundamental groups of non-compact arithmetic hyperbolic manifolds contain thin surface subgroups.
New characteristic classes for manifold bundles with specific forms.
problem Characteristic classes for manifold bundles with indefinite intersection forms.
method Using a technique of Millson--Raghunathan, cohomology for non-uniform arithmetic lattices is produced.
result New characteristic classes defined on a finite cover of BDiff(M), nontrivial for specific manifolds. Develops Kleinian Sphere Packings and Bugs, proving their arithmetic origins.
problem Understanding sphere packings and their arithmetic origins in various dimensions.
method Introduces Kleinian Sphere Packings and Bugs, extending Arithmeticity Theorem.
result Kleinian packings and Bugs come from Q-arithmetic lattices of simplest type.
New lattices in higher rank contain a fixed 3-manifold group with increasing systole.
problem Finding lattices with a fixed 3-manifold group and large systole.
method Constructing arithmetic lattices in SL(8,R) with specific properties. result Existence of lattices with large systole containing a fixed 3-manifold group.
Study jigsaw constructions of hyperbolic lattices and answer questions on arithmeticity and pseudomodularity.
problem Constructing and analyzing non-commensurable, non-uniform, non-arithmetic lattices in hyperbolic geometry.
method Hyperbolic jigsaw construction and recursive formulas for tessellations.
result Demonstration of recursive formula for tessellation of hyperbolic plane, generalizing Farey addition.
We study the covolumes of arithmetic lattices in PSL2(R)n for n≥2 and identify uniform and non-uniform irreducible lattices of minimal covolume. More precisely, let μ be the Euler-Poincaré measure on PSL2(R)n and χ=μ/2n. We show that the Hilbert modular group $PSL_2(\mathfrak o_{k_{49…
We discuss the geometry of some arithmetic orbifolds locally isometric to a product of real hyperbolic spaces of dimension two and three, and prove that certain sequences of non-uniform orbifolds are convergent to this space in a geometric ("Benjamini--Schramm") sense for hyperbolic three--space and a product of hyperb…
Arithmetic Kleinian groups are distinguished by their finite quotients.
problem Distinguishing arithmetic Kleinian groups among all finitely generated residually finite groups.
method Constructing specific examples of arithmetic Kleinian groups and proving their profinite rigidity.
result Arithmetic Kleinian groups are uniquely identified by their finite quotients.
New rigidity theorem for product of lattices.
problem Understanding quasi-isometry of product lattices.
method Demonstrated rigidity for product of non-uniform rank one lattice and nilpotent lattice.
result Any quasi-isometric group is an extension of a non-uniform rank one lattice by a nilpotent lattice.
New non-arithmetic lattice found in PU(3,1)
problem Arithmeticity of Couwenberg-Heckman-Looijenga lattices
method Study of arithmeticity and non-arithmetic lattices in PU(n,1)
result Found a non-arithmetic lattice in PU(3,1) not commensurable to Deligne-Mostow lattice
Hybrid subgroups found in non-arithmetic PU(2,1) lattices.
problem Exploring hybrid subgroups in non-arithmetic PU(2,1) lattices.
method Exploring hybrid subgroups of certain non-arithmetic lattices in PU(2,1). Showing that Mostow's lattices are virtually hybrids and some are hybrids of two non-commensurable arithmetic lattices in PU(1,1).
result Mostow's lattices are virtually hybrids and some are hybrids of two non-commensurable arithmetic lattices in PU(1,1).
Bounds on homology of hyperbolic orbifolds using simplicial models.
problem Bounding homology of hyperbolic orbifolds.
method Efficient simplicial model of the thick part of hyperbolic orbifolds.
result Linear bounds on Betti numbers and torsion homology in terms of volume.
Course on arithmetic lattices at EPFL.
problem Understanding arithmetic lattices.
method Introductory course on arithmetic lattices.
result Introduction to arithmetic lattices.
Study on curvature properties and Shafarevich conjecture for complex hyperbolic manifolds.
problem Existence and nonexistence of Kähler metrics with nonpositive curvature on toroidal compactifications.
method Analysis of toroidal compactifications of finite volume complex hyperbolic manifolds, verification of Shafarevich conjecture.
result Verification of Shafarevich conjecture for compactifications of quotients of complex hyperbolic space by non-uniform arithmetic lattices.
New property identifies arithmetic lattices from nonuniform lattices.
problem Characterizing arithmetic lattices among nonuniform lattices.
method Introduced Bounded Clustering (B-C) property.
result B-C property uniquely identifies arithmetic lattices.
We show that the non-arithmetic lattices in PO(n,1) of Belolipetsky and Thomson (2011), obtained as fundamental groups of closed hyperbolic manifolds with short systole, are quasi-arithmetic in the sense of Vinberg, and, by contrast, the well-known non-arithmetic lattices of Gromov and Piatetski-Shapiro are not quasi-a…
New thin subgroups found in special linear groups via bending techniques.
problem Finding thin subgroups of lattices in special linear groups.
method Techniques from convex projective geometry.
result Infinitely many non-commensurable lattices with thin subgroups.
New research shows certain arithmetic lattices can't be LERF.
problem Determining if arithmetic lattices are LERF.
method Analyzing trialitarian arithmetic lattices in PSO7,1(R). result Trialitarian arithmetic lattices in PSO7,1(R) are not LERF. Proves properties of arithmetic lattices and hyperbolic manifolds.
problem Properties of arithmetic lattices and hyperbolic manifolds.
method Study of normalizers of lattices and subgroup growth theory.
result Every arithmetic lattice has the property of being the normalizer of many sublattices.
Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
problem Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
method Proving non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
result Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
Study answers arithmeticity question for normal subgroup of lattices.
problem Arithmeticity of discrete subgroups of semisimple Lie groups with dense commensurators.
method Examined normal subgroups of lattices in semisimple Lie groups.
result Positive answer to Greenberg-Shalom's question for lattices.
We prove, under the assumption of the virtual fibration conjecture for arithmetic hyperbolic 3-manifolds, that all arithmetic lattices in O(n,1), n> 4, and different from 7, are non-coherent. We also establish noncoherence of uniform arithmetic lattices of the simplest type in SU(n,1), n> 1, and of uniform lattices in …
Study C-Fuchsian subgroups of non-arithmetic lattices.
problem Understand structure and fundamental domains of C-Fuchsian subgroups. method General procedure to analyze structure and show fundamental domains lie on a complex geodesic.
result Fundamental domains of C-Fuchsian subgroups lie on a complex geodesic homeomorphic to the unit disk. Let X=S×E×B be the metric product of a symmetric space S of noncompact type, a Euclidean space E and a product B of Euclidean buildings. Let Γ be a discrete group acting isometrically and cocompactly on X. We determine a family of quasi-isometry invariants for such Γ, namely the k-dimension…
New complex hyperbolic lattices discovered from triangle groups.
problem Finding new non-arithmetic complex hyperbolic lattices.
method General procedure to produce fundamental domains for complex hyperbolic triangle groups.
result Some triangle groups yield new commensurability classes, increasing the count to 22.
We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of righ…
The study introduces pseudo-arithmeticity for certain lattices in hyperbolic spaces.
problem Understanding the structure of certain lattices in hyperbolic spaces.
method Introducing pseudo-arithmeticity and showing covolumes relate to special values of L-functions.
result Covolumes of certain lattices correspond to rational linear combinations of special values of L-functions.
New lattices are linked to higher hypergeometric functions.
problem Understanding non-arithmetic lattices in PU(2,1).
method Showed all known non-arithmetic lattices are monodromy groups of higher hypergeometric functions.
result Non-arithmetic lattices in PU(2,1) are linked to higher hypergeometric functions.
We prove that if G is a non-uniform lattice in a rank-one semi-simple Lie group $\ne Isom(\H^2_\R)$ then G is quasi-isometrically co-Hopf. This means that every quasi-isometric embedding G→G is coarsely onto and thus is a quasi-isometry.
The paper proves Zimmer's conjecture for non-uniform lattices by controlling mass escape and Lyapunov exponents.
problem Proving Zimmer's conjecture for non-uniform lattices in higher-rank semisimple Lie groups.
method Establishes finiteness of low-dimensional actions, introduces novel techniques to control mass escape and Lyapunov exponents.
result Proves Zimmer's conjecture for many non-uniform lattices, improving previous results.
In this note, we study deformations of a non-uniform real hyperbolic lattice in quaternionic hyperbolic spaces. Specially we show that the representations of the fundamental group of the figure eight knot complement into PU(2,1) cannot be deformed in PSp(2,1) out of PU(2,1) up to conjugacy.
Study of complex projective manifolds using arithmetic lattices.
problem Holomorphic convexity for toroidal compactifications of ball quotients.
method Show that Albanese mapping on an étale covering space generates jets on the interior.
result Shafarevich conjecture on holomorphic convexity satisfied in dimension 2 for arithmetic lattices.
This paper studies the covolumes of nonuniform arithmetic lattices in PU(n, 1). We determine the smallest covolume nonuniform arithmetic lattices for each n, the number of minimal covolume lattices for each n, and study the growth of the minimal covolume as n varies. In particular, there is a unique lattice (up to conj…
Computes presentations for specific arithmetic hyperbolic lattices.
problem Computing presentations for cusped arithmetic hyperbolic lattices.
method Applying Macbeath's classical result to invariant horoball covers.
result Computations for specific groups like Picard modular and quaternion hyperbolic.
Thin groups found in specific lattices.
problem Embedding right-angled Coxeter groups in arithmetic lattices.
method Using Agol's unpublished argument, embedding in indefinite orthogonal groups.
result Irreducible right-angled Coxeter groups embed as thin subgroups.
Finite actions of lattices on manifolds proven for certain groups.
problem Finite actions of lattices on compact manifolds.
method Uses machinery from Brown, Fisher, and Hurtado.
result Finite actions proven for lattices in p-adic and S-arithmetic groups. New lattices in higher dimensions have dense surface subgroups.
problem Finding dense subgroups in higher-dimensional arithmetic lattices.
method Exhibited nonuniform arithmetic lattices in SO(n,1).
result Contain Zariski-dense surface subgroups.
Study complex hyperbolic lattices and their relation to strict hyperbolization.
problem Understanding the relationship between complex hyperbolic lattices and strict hyperbolization.
method Analyzing the fundamental groups of complex hyperbolic manifolds and spaces arising from strict hyperbolization.
result Uniform lattices in PU(n,1) cannot be fundamental groups of Charney-Davis strict hyperbolizations when n ≥ 2.
We show that S-arithmetic lattices in semisimple Lie groups with no rank one factors are quasi-isometrically rigid.
The principle result of this article is the determination of the possible finite subgroups of arithmetic lattices in U(2,1).
The paper explores subspaces in hyperbolic lattices and their arithmetic properties.
problem Arithmeticity criterion for hyperbolic lattices and suborbifolds.
method Analysis of totally geodesic suborbifolds and Vinberg's commensurability invariants.
result Arithmeticity of hyperbolic orbifolds is linked to the existence of infinitely many fc-subspaces.
Minimal crossing number found in arithmetic curve systems.
problem Finding the minimal crossing number in arithmetic curve systems.
method Analyzing systoles of hyperbolic surfaces associated with congruence lattices in SL2(Z).
result Minimal crossing number is asymptotically achieved.
Proof shows volumes of certain geometric representations are always integers.
problem Integrality of volumes of specific geometric representations.
method Elementary, combinatorial-geometrical proof.
result Volumes of representations are integers when n≥2. New bounds on diameters and generators for specific lattices and graphs.
problem Finding bounds on diameters and generators for arithmetic lattices and Ramanujan graphs.
method Analyzing arithmetic lattices from Eichler orders in quaternion algebras, applying techniques to definite quaternion algebras.
result Bounds on diameters and generators for arithmetic lattices and Ramanujan graphs.
New non-arithmetic ball quotients from elliptic curves on Abelian surfaces.
problem Constructing non-arithmetic ball quotients from Abelian surfaces.
method Branched covers of Abelian surface quotients by finite groups.
result Alternative construction of lattices from elliptic curves.
The paper improves Vinberg's algorithm for arithmetic hyperbolic lattices.
problem Finding maximal reflection sublattices in arithmetic hyperbolic lattices.
method Provided an effective termination condition for Vinberg's semi-algorithm.
result The algorithm becomes an effective method for finding maximal reflection sublattices.
We prove that an irreducible lattice in a semisimple algebraic group is virtually isomorphic to an arithmetic lattice if and only if it admits a faithful self-similar action on a rooted tree of finite valency.